Here, \(\beta \) , \(\eta \) , and \(\zeta \)  are three physical quantities reflecting the relative strength of local magnetism, for which the ratio of exchange splitting on sublattices \(A_1\) , \(B_1\) , \(A_2\)  and \(B_2\)  is \(1:\beta :\eta :\zeta \eta \) . By setting \(\beta =\pm 1\)  and \(\zeta =\pm 1\) , we assume for simplicity that the absolute values of magnetization in both sublatices of each monolayer of BLG are the same. The amplitude difference of magnetization between two opposite layers depends on the value of \(\eta \) . Because the proximity effect is short-range, \(\beta \)  and \(\eta \)  are two completely independent parameters. The index \(\eta ^{\prime }\)  depends on the coupling details between BLG and its proximity partners [1]}, [2]}, [3]}, [4]}. In Table REF , we show the magnetic type in the parameter space \(\lbrace \beta ,\eta ,\zeta \rbrace \) . Depending on the parameter values, the magnetism types of FM, AFM, FIM are distinguished. To avoid the complexity caused by too many parameters in the model (REF ), we restrict ourselves below to the qualitative discussion of the effects of the spin-orbit coupling perturbation \(\cal H^{\prime }\) .
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